https://file.aiquickdraw.com/mathbot/file/1726745860493a0blpa9h.jpg

The image asks to evaluate two expressions involving combinations and permutations:

1. \( _8C_4 \) (combinations)
2. \( _9P_3 \) (permutations)

### 1. Evaluating \( _8C_4 \):
The combination formula is given by:
\[
_8C_4 = \frac{8!}{4!(8 - 4)!} = \frac{8!}{4!4!}
\]
Simplifying this:
\[
_8C_4 = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70
\]

### 2. Evaluating \( _9P_3 \):
The permutation formula is given by:
\[
_9P_3 = \frac{9!}{(9 - 3)!} = \frac{9!}{6!}
\]
Simplifying this:
\[
_9P_3 = 9 \times 8 \times 7 = 504
\]

### Final Answers:
- \( _8C_4 = 70 \)
- \( _9P_3 = 504 \)

Would you like any further clarification or details?

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Here are 5 questions to expand your understanding:
1. What is the difference between a combination and a permutation?
2. How would \( _9C_3 \) differ from \( _9P_3 \) in terms of formula and result?
3. How does the factorial function play a role in these formulas?
4. In what types of problems would you use permutations instead of combinations?
5. What happens to the values of permutations and combinations if the numbers are equal (e.g., \( _5C_5 \) or \( _5P_5 \))?

**Tip**: Always remember that order matters in permutations but not in combinations!

Learn how to evaluate combinations and permutations, including solving _8C_4 (combinations) and _9P_3 (permutations). Understand the use of factorials in combinatorics. Suitable for students in Grades 9-12.

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